LINEAR ALGEBRA GUIDE
Matrix methods for solving linear systems
A linear system can be written as A·x = b, where A contains coefficients, x contains unknown values, and b contains constants. This guide explains when Gaussian elimination, Gauss-Jordan RREF, the inverse method, rank checks, pivot columns, row-space reasoning, and Cramer’s rule are useful.Explore algebra & linear systems →FORMULA OR CORE IDEA
A·x = bWORKED EXAMPLE
For 2x + y = 5 and x − y = 1, set A = [[2, 1], [1, −1]], x = [x, y], and b = [5, 1]. Solving the system gives x = 2 and y = 1.STEP BY STEP
Translate equations into A, x, and b
Keep variables in the same order in each equation. The coefficient matrix is A, the unknown column is x, and the constants become b. This structure makes the same system usable in several matrix methods.
Use Gaussian elimination for an efficient general method
Gaussian elimination reduces an augmented matrix to REF and then uses back-substitution. It is a strong default for many square systems and is practical for larger systems than Cramer’s rule.
Use Gauss-Jordan RREF when you want a readable reduced form
RREF continues row reduction until each pivot is 1 and the other entries in pivot columns are 0. This makes a unique solution easy to read, while also exposing free variables, dependent rows, row-space structure, or inconsistent rows.
Use the inverse method only when det(A) is not zero
The inverse method uses x = A⁻¹b. It requires a square, non-singular matrix, so it is most useful when you also want the inverse or when the same A matrix is paired with more than one b vector.
Use Cramer’s rule for small square systems
Cramer’s rule replaces one coefficient column at a time with b and compares determinants. It is useful for learning and small systems, but it is usually less efficient than row reduction as the system grows.
Use determinant and rank to interpret special cases
A zero determinant prevents an inverse, but RREF and rank are needed to decide whether the system has no solution or infinitely many solutions. Do not assume every singular system is inconsistent. Compare rank(A) with the rank of the augmented matrix when classifying a system.
Use rank to understand special systems
If every variable column has a pivot and the augmented system is consistent, the system has a unique solution. Missing pivots usually mean free variables. A row such as [0 0 0 | 5] means the equations contradict each other.
Verify the final vector against the original equations
Substitute the solved values back into the original equations, especially when values are rounded or come from measured data. A quick check confirms that A·x matches b.